Gorontalo Utara

Boolean Algebra Expression Laws, Rules, Theorems and Solved Examples

Published

3 years ago

18/08/2022

There are some set of logical expressions which we accept as true and upon which we can build a set of useful theorems. These sets of logical expressions are known as Axioms or postulates of Boolean Algebra. An axiom is nothing more than the definition of three basic logic operations (AND, OR and NOT). All axioms defined in boolean algebra are the results of an operation that is performed by a logical gate.

In contrast, in a list of some but not all of the same laws, there could have been Boolean laws that did not follow from those on the list, and moreover there would have been models of the listed laws that were not Boolean algebras. These generalized expressions are very important as they are used to simplify many Boolean Functions and expressions. Minimizing the boolean function is useful in eliminating variables and Gate Level Minimization. The theorem states that the complement of the OR operation between two or more variables is equivalent to the AND operation of their complements.

\(B\) is
unique up to \(A\)-isomorphism, and is called the completion of
\(A\). If \(f\) is a homomorphism from a BA \(A\) into a complete BA
\(B\), and if \(A\) is a subalgebra https://1investing.in/ of \(C\), then \(f\) can be
extended to a homomorphism of \(C\) into \(B\). Another general algebraic notion
which applies to Boolean algebras is the notion of a free
algebra.

Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such as set. In electrical and electronic circuits, Boolean algebra is used to simplify and analyze the logical or digital circuits. The other theorems in Boolean algebra are complementary theorem, duality theorem, transposition theorem, redundancy theorem and so on. All these theorems are used to simplify the given Boolean expression.

Furthermore, Boolean algebras can then be defined as the models of these axioms as treated in § Boolean algebras. The equivalent logical operators to these operations are given below. Boolean algebra is a type of algebra that is created by operating the binary system. In the year 1854, George Boole, an English mathematician, proposed this algebra. This is a variant of Aristotle’s propositional logic that uses the symbols 0 and 1, or True and False.

Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method. Embark on a transformative journey towards GATE success by choosing Data Science & AI as your second paper choice with our specialized course. If you find yourself lost in the vast landscape of the GATE syllabus, our program is the compass you need. Boolean Algebra also called Logical Algebra is a branch of mathematics that deals with Boolean Varaibles such as, 0 and 1. Of course, it is possible to code more than two symbols in any given medium.

The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 (some authors use the opposite convention).
However, we could put a circle for x in those boxes, in which case each would denote a function of one argument, x, which returns the same value independently of x, called a constant function.
This result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice.
At times, it is not even clear which collection of axioms a proof appeals to.

The system has at least two different models – one is the natural numbers (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the cardinality of the continuum). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, axiomatic definition of boolean algebra the property distinguishing these models is their cardinality — a property which cannot be defined within the system. A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system.

Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master’s thesis, A Symbolic Analysis of Relay and Switching Circuits. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port. When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, the members of each pair are called dual to each other. The duality principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged. Writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them.

Commutative Law

Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively. It is used to analyze and simplify digital circuits or digital gates. It has been fundamental in the development of digital electronics and is provided for in all modern programming languages. The term “Boolean algebra” honors George Boole (1815–1864), a self-educated English mathematician. Boole’s formulation differs from that described above in some important respects.

Complementation Laws

This describes the scenario where the undefined terms of a first axiom system are provided definitions from a second, such that the axioms of the first are theorems of the second. The second law states that the complement of the sum of variables is equal to the product of their individual complements of a variable. The first law states that the complement of the product of the variables is equal to the sum of their individual complements of a variable. This means that if you want to find the complement of the OR operation of two or more variables, you can take the complement of each variable individually and then use the AND operation between their complements. This means that if you want to find the complement of the AND operation of two or more variables, you can take the complement of each variable individually and then use the OR operation between their complements. There are two basic theorems of great importance in Boolean Algebra, which are De Morgan’s First Laws, and De Morgan’s Second Laws.

Boolean algebra (structure)

Instead of showing that the Boolean laws are satisfied, we can instead postulate a set X, two binary operations on X, and one unary operation, and require that those operations satisfy the laws of Boolean algebra. The elements of X need not be bit vectors or subsets but can be anything at all. The closely related model of computation known as a Boolean circuit relates time complexity (of an algorithm) to circuit complexity. The theory of Boolean algebras was founded in 1847 by Boole, who considered it a form of ‘calculus’ adequate for the study of logic.

Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff’s 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.

For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies. A truth table represents all the combinations of input values and outputs in a tabular manner. All the possibilities of the input and output are shown in it and hence the name truth table. In logic problems, truth tables are commonly used to represent various cases.

OR Laws

The theorem states that the complement of the AND operation between two or more variables is equivalent to the OR operation of their complements. It is used to simplify logical circuits that are the backbone of modern technology. The inverse of the boolean variable is called the complement of the variable. A function of the Boolean Algebra that is formed by the use of Boolean variables and Boolean operators is called the Boolean function.

Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way.[14][15][16] Boolean algebra is not sufficient to capture logic formulas using quantifiers, like those from first order logic. The two important theorems which are extremely used in Boolean algebra are De Morgan’s First law and De Morgan’s second law.

Using an axiomatic proof (i.e. using only the basic axioms and theorems of Boolean algebra).However, no matter what I do, I can’t seem to get things to line up correctly. Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory. That is, up to isomorphism, abstract and concrete Boolean algebras are the same thing. This result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice. This strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability.

Gorontalo Utara

Pakar Siber: “Yang Ditangkap Bukan Bjorka Asli”

Published

3 weeks ago

04/10/2025

Redaksi Barakati

Penangkapan hacker kontroversial Bjorka oleh Polda Metro Jaya pada Kamis (2/10/2025) terus menuai keraguan warganet. Meski polisi mengumumkan penangkapan WFT (22), akun Instagram yang diklaim milik Bjorka masih aktif membantah dan bahkan membocorkan data Badan Gizi Nasional. Reaksi warganet di X (Twitter) pun langsung membanjiri linimasa.

“Ketika Bjorka up story IG, lalu siapa yang ditangkap???” tanya akun @Opposisi6890, mendapatkan ratusan like dan repost. Tidak sedikit yang menganggap penangkapan ini sekadar pengalihan isu. “@baratieee_ menulis, ‘Soal hengker bjorka yang ketangkap itu, filling gw sih cuman buat pengalihan isu. Yakin gw bukan hengker bjorka asli itu.'” Sementara, @yusabdul menyoroti, “Bjorka yang sesungguhnya adalah orang dalam yang berani bayar ke pemilik server database instansi/perusahaan, termasuk Dukcapil. Gak mungkin bocah umur belasan tahun.”

Pakarnya, Teguh Aprianto, pendiri Ethical Hacker Indonesia, juga angkat suara, “Polisi dengan pedenya bilang kalau mereka nangkap Bjorka terus konpers seakan-akan yang ditangkap itu kasus yang wah banget. Padahal yang ditangkap itu cuma bocah yang selama ini ngaku-ngaku jadi Bjorka dan bocah yang suka repost thread orang lain.”

Penangkapan berawal dari laporan bank swasta tentang pembocoran data 4,9 juta akun nasabah yang diunggah akun X @bjorkanesiaa. “Peran dari tersangka, yang bersangkutan adalah pemilik akun media sosial X dengan nama Bjorka dan @bjorkanesiaa,” jelas AKBP Reonald Simanjuntak, Kasubbid Penmas Bidhumas Polda Metro Jaya.

Menurut AKBP Fian Yunus, penyelidikan terhadap WFT telah berjalan enam bulan. “Pelaku ini bermain di dark web sejak 2020, mengeksplor berbagai forum gelap tempat jual beli data,” ungkapnya.

AKBP Herman Edco menambahkan, “Selain data bank, WFT juga diduga memperoleh data ilegal dari sektor kesehatan dan perusahaan swasta di Indonesia. Data-data itu dijual di media sosial dengan harga mencapai puluhan juta rupiah. Motif pelaku adalah pemerasan, meski belum sempat terjadi. Barang bukti berupa komputer dan ponsel yang digunakan sudah diamankan.”

WFT kini dijerat Pasal 46 jo Pasal 30 dan/atau Pasal 48 jo Pasal 32 dan/atau Pasal 51 ayat (1) jo Pasal 35 UU ITE, dengan ancaman hingga 12 tahun penjara.

Namun, pihak kepolisian sendiri belum memastikan apakah WFT adalah Bjorka asli yang kerap membocorkan data pemerintah sejak 2022. “Everybody can be anybody on the internet,” kata AKBP Fian Yunus.

Kasus ini mengingatkan pada penangkapan serupa sebelumnya yang juga menimbulkan keraguan publik. Sebuah sumber internasional, The Jakarta Post, menulis bahwa identitas Bjorka tetap sulit dipastikan dan bahwa “identitas pelaku yang sebenarnya belum terkonfirmasi karena siapapun bisa mengatasnamakan Bjorka di internet”. banyak yang menyoroti aktivitas Bjorka di dark web sejak 2020 dan ancaman pidana maksimal yang kini dihadapinya.

Gorontalo Utara

Netizen Heboh! Bangunan mirip toilet di Boyolali telan anggaran 112 juta

Published

1 month ago

22/09/2025

Redaksi Barakati

Sebuah bangunan kecil di pinggir area persawahan Kecamatan Ngemplak, Kabupaten Boyolali, baru-baru ini menjadi sorotan publik setelah diketahui dibangun dengan anggaran Rp 112,8 juta. Bangunan berukuran sekitar 1,5 x 1,5 meter ini memiliki dinding bata yang telah diplester dan diaci rapi, namun belum dicat. Ventilasi menggunakan roster, atap dari galvalum, dan pintu terbuat dari triplek lapis seng. Halaman depan dilapisi rabat beton agar terlihat lebih rapi.

Wildan, kontraktor dari Rebwild Construction, menilai bahwa biaya sebesar Rp 25 juta sudah cukup untuk membangun bangunan serupa, termasuk fondasi bawah dan pemasangan bor sumur. “Rp 25 juta juga cukup, mahal malah. Fondasi bawahnya pun sudah, kalau sama bor sumurnya masih masuk kayaknya, Rp 25 juta sudah sampai terbangun,” jelas Wildan.

Namun, anggaran Rp 112,8 juta tersebut mencakup keseluruhan paket kegiatan irigasi perpompaan yang memiliki manfaat langsung bagi petani. Sekretaris Dinas Pertanian Boyolali, Retno Nawangtari, menjelaskan bahwa anggaran tersebut termasuk pembuatan sumur bor, pembelian mesin pompa, pemasangan pipa, instalasi listrik, dan pembangunan rumah pompa. “Paling banyak anggaran untuk pembuatan sumur dalam,” kata Retno.

Kepala Dinas Pertanian Kabupaten Boyolali, Suyanta, menambahkan bahwa kegiatan pembangunan irigasi perpompaan ini dilakukan secara swakelola oleh kelompok tani penerima manfaat. “Perlu ditegaskan bahwa kegiatan Irpom Tahun 2024 ini bukan dilaksanakan langsung oleh Dinas (Pertanian), melainkan melalui mekanisme swakelola oleh kelompok tani penerima manfaat, sesuai ketentuan peraturan perundang-undangan,” katanya.

Lokasi bangunan kecil ini berada di pinggir areal persawahan wilayah Desa Gagaksipat, Kecamatan Ngemplak, Kabupaten Boyolali. Tepatnya di sebelah utara landasan pacu Bandara Adi Soemarmo. Bangunan ini cukup mudah ditemukan karena berada persis di pinggir jalan raya ruas Mangu-Donohudan yang menghubungkan Bandara dan Asrama Haji Donohudan.

Gorontalo Utara

Bongkar! Mahfud MD Jelaskan Solusi Pemerintahan Agar Roda Negara Kembali Lancar

Published

2 months ago

02/09/2025

Redaksi Barakati

Jakarta – Prof. Mahfud MD mengulas secara mendalam situasi demonstrasi yang sempat mencekam di berbagai kota Indonesia pada akhir Agustus 2025. Mahfud MD menegaskan bahwa walaupun kekerasan telah berhasil diredam terutama berkat langkah tegas Presiden Prabowo, masalah mendasar yang menjadi pemicu demonstrasi belum terselesaikan.

Demo yang awalnya dipicu oleh kebijakan pemerintah ini melahirkan kerusuhan hebat, termasuk pembakaran gedung DPR, korban jiwa, dan kerusakan harta benda. Situasi mulai membaik sejak Minggu malam dengan belum ada demonstrasi besar menggantikan.

Masalah utama yang belum dijawab adalah akumulasi berbagai persoalan sosial dan ekonomi, seperti tingginya angka PHK dan pengangguran, serta persoalan pajak dan pungutan yang memicu ketidakpuasan masyarakat. Penegakan hukum yang lemah, praktik kriminalisasi, politisasi hukum, serta kasus korupsi yang tidak jelas penyelesaiannya semakin memperparah kepercayaan publik terhadap pemerintah.

dikutip dari podcast Terus Terang bersama Mahfud MD, dia menyatakan bahwa penegakan hukum yang masih kacau membuat sulitnya mencari investor karena reputasi hukum yang buruk. Ia juga menyoroti peran ormas Islam yang dianggap telah jauh dari rakyat dan terlalu dekat dengan pemerintah sehingga tidak menjalankan perannya sebagai pemandu moral masyarakat secara tepat.

Pentingnya reformasi KPK dan sinergi aparatur negara dalam penegakan hukum juga menjadi sorotan utama untuk membangun pemerintahan yang profesional dan bersih. Selain itu, Mahfud MD mengkritik kabinet yang dianggap terlalu besar dan banyak pejabat bermasalah hukum, sehingga melemahkan kerja pemerintah.

Mahfud MD mengingatkan pentingnya kepemimpinan yang berani menerima kritik jujur dan penegakan hukum yang tegas agar negara ini bisa selamat dari masalah panjang yang melekat dalam pemerintahan.

Dalam kesempatan itu, Mahfud MD berbagi pengalamannya berani menyampaikan kritik langsung kepada Presiden Jokowi terkait sejumlah kasus besar seperti BLBI, menunjukkan pentingnya keberanian menyuarakan kebenaran demi kebaikan bangsa.

Ia mengharapkan Presiden Prabowo dapat mengambil langkah cepat menyelesaikan masalah hukum dan evaluasi kabinet untuk memenuhi aspirasi masyarakat agar roda pemerintahan kembali berjalan efektif.

Selain itu, Mahfud MD juga menanggapi sikap pemerintah terhadap demonstrasi, menegaskan bahwa TNI dan Polri harus bertindak tegas sesuai hukum namun tetap menghormati kebebasan berpendapat di Indonesia.